Document Type : Research Paper

**Authors**

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.

**Abstract**

It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equation. The fractional derivatives are described based on the Caputo sense. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. In this work, the Chebyshev cardinal functions together with the Chebyshev cardinal operational matrix of fractional derivatives are used for numerical solution of a class of fractional differential equations. The main advantage of this approach is that it reduces fractional problems to a system of algebraic equations. The method is applied to solve nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

**Keywords**

- Fractional-order differential equation
- operational matrix of fractional derivative
- Caputo derivative
- Chebyshev cardinal function
- collocation method

**Main Subjects**

Q. M. Al-Mdallal, M. I. Syam and M. N Anwar, A collocation-shooting method for solving fractional boundary value problems, Commun. Nonlinear Sci. Numer. Simul. **15** (2010), no. 12, 3814--3822.

Z. B. Bai and H. S. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. **311** (2005), no. 2, 495--505.

H. Bbeyer and S. Kempe, Definition of physically consistent damping laws with fractional derivatives, Z. Angew. Math. Mech. **75** (1995), no. 8, 623--635.

L. Blank, Numerical treatment of differential equations of fractional order, Nonlinear World 4 (1997), no. 4, 473--491

J. P. Boyd, The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions, Appl. Math. Comput. **29** (1989), no. 1, part I, 49--67.

J. P. Boyd, Polynomial series versus sinc expansions for functions with corner or endpoint singularities, J. Comput. Phys. **64** (1986), no. 1, 266--270.

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Inc., Mineola, 2001.

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlger Wien, New York, 1997.

Y. Censiz, Y. Keskin and A. Kurnaz, The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, J. Franklin Inst. **347** (2010), no. 2, 452--466.

V. Daftardar-Geiji and H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl. **301** (2005), no. 2, 508--18.

V. Daftardar-Geiji and H. Jafari, Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. Math. Anal. Appl. **328** (2007), no. 2, 1026--1033.

M. Dehghan and M. Lakestani, The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation, Numer. Methods Partial Differential Equations **25** (2009), no. 4, 931--938.

M. Dehghan, J. Manafian and A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Zeitschrift fur Naturforschung, J. Phys. Sci. **65** (2010), no. 11, 935--949.

W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equationd, Nonlinear Anal. **72** (2010), no. 3-4, 1768--1777.

W. Gautschi, The Condition of Polynomials in Power Form, Math. Comp. **33** (1979), no. 145, 343--352

I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 3, 674--684.

S. Irandoust--pakchin, Exact solutions for some of the fractional differential equations by using modification of He's variational iteration method, Math. Sci. Q. J. **5** (2011), no. 1, 51--60.

S. Irandoust-pakchin, M. Dehghan, S. Abdi-mazraeh and M. Lakestani, Numerical solution for a class of fractional convection-diffusion equation using the flatlet oblique multiwavelets, J. Vib. Control **20** (2014), no. 6, 913--924.

S. Irandoust-pakchin, H. Kheiri and S. Abdi-Mazraeh, Chebyshev cardinal functions: an effective tool for solving nonlinear Volterra and Fredholm integro-differential equations of fractional order, Iran. J. Sci. Technol. Trans. A Sci. **37** (2013), no. 1, 53--62.

S. Irandoust-pakchin, H. Kheiri and S. Abdi-Mazraeh, Efficient computational algorithms for solving one class of fractional boundary value problems, Comput. Math. Math. Phys. **53** (2013), no. 7, 920--932.

V. Lakshmikantham and A. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. **21** (2008), no. 8, 828--834.

M. Lakestani, M. Dehghan and S. Irandoust--pakchin, The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul. **17** (2012), no. 3, 1149--1162

M. Lakestani and M. Dehghan, Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions, Comput. Phys. Commun. **181** (2010), no. 5, 957--966.

M. Lakestani and M. Dehghan, The use of Chebyshev cardinal functions for the solution of a partial differential equation With an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math. 235 (2010), no. 3, 669--678.

M. Lakestani and M. Dehghan, Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions, Int. J. Comput. Math. **87** (2010), no. 6, 1389--1394.

M. Lia, S. Jimenezc, N. Niea, Y. Tanga and L. Vazqueze, Solving Two--point boundary value problems of fractional differetial equations by spline collocation methods, Available from http://www.cc.ac.cn/2009 researth-report/0903.pdf, (2009), 1--10.

V. E. Lynch, B. A. Carreras, D. Del-Castillo-Negrete, K. M. Ferriera-Mejias and H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. **192** (2003), no. 2, 406--421.

F. Mainardi, Fractional relaxiation and fractional diffusion equations: mathematical aspects, Proceedings of the 14th IMACS World Congress, **1** (1994) 329--32.

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996), 291--348, CISM Courses and Lectures, 378, Springer, Vienna, 1997.

[30] M. M. Meerschaert, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. **56** (2006), no. 1, 80--90.

M. U. Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. **16** (2011), no. 11, 4163--4173.

M. Ochmann and S. Makarov, Representation of the absorption of nonlinear waves by fractional derivatives, J. Acoust. Soc. Amer. **94** (1993), no. 6, 2--9.

Z. Odibat, S. Momani and H. Xu, A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. Math. Model. **34** (2010), no. 3, 593--600.

Z. Odibat and S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J.

G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, Fifth edition, Academic Press, Inc., 1980.

I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, 1999.

E. A. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math. Comput. **176** (2006), no. 1, 1--6.

Z. Shuqin, Existence of solution for a boundary value problem of fractional order, Acta Math. Sci. Ser. B Engl. Ed. **26** (2006), no. 2, 220--228.

C. Tadjeran and M. M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys. **220** (2007), no. 2, 813-- 823.

S. A. El-Wakil, A. Elhanbaly and M. A. Abdou, Adomian decomposition method for solving fractional nonlinear differential equations, Appl. Math. Comput. **182** (2006), no. 1, 313--324.

September and October 2016

Pages 1107-1126

**Receive Date:**10 January 2015**Revise Date:**05 July 2015**Accept Date:**05 July 2015